SAVE YOUR MANA!!! By dipping into the Offensive Tree instead of the Utility Tree for the remaining 9 Mastery points, you will run out of Mana if you don't recall after a few ganks and clears.

Starting Items : Hunter's Machete + 5 Health Potion
Starting Skill : Despair
First Clear Time Order: Wolves -> Blue Golem -> Wraiths -> Wolves -> Red Lizard -> Wraiths -> GANK!
Purchase on First Recall : Boots + Philosopher's Stone

Level 1-3 =
Level 4-5 =
Level 6+ =
This is basically what you do...
-> -> -> ->
Bandage Toss -> AutoAttack -> Despair -> Tantrum -> Curse of the Sad Mummy
Landing your Cursed Touch Passive WILL MAKE A DIFFERENCE

Don't use your Ultimate unless you really need to. There WILL be times where the enemy would've died if your ultimate wasn't on Cooldown.

Starting Skill: Despair
Starting Items: Hunter's Machete and 5x Health Potion

There are 2 routes you can use but with some suggestions and testing the best route is...

Wolves -> Blue Golem -> Wraiths -> Wolves -> Red Lizard -> Wraiths -> Level Up Bandage Toss -> GANK -> Continue Jungling or recall if you're low.

NOTE: Try to get 1050 gold then recall for both Boots and Philosopher's Stone

Don't forget to ask your teammates to help you with Wolves and Blue Golem. They will save you Time and Health

GANK MID LANE A LOT! YOUR PASSIVE WILL AMPLIFY YOU AND YOUR MID LANER'S DAMAGE.

NOTE: Although getting the kill from the gank (if successful) would really really speed up your build, it's still better to give it to your Teammate if possible so they have a higher chance to win the Lane. They need it more than you do.

NOTE: Bandage Toss uses a lot of Mana, save it for ganks and not on the jungle monsters even though it's very tempting. It is also possible to use it for escaping by using Bandage Toss over a wall and grabbing onto a Jungle Monster, pulling yourself over the wall. Please check the mini map first before you toss as your most likely gonna die if u grabbed nothing.

Basically from now on, just clear your Jungle before you gank or if you see any lanes being pushed, try to gank those FIRST.

You could also invade and take your enemy's Jungle Monsters.
I RECOMMEND stealing the enemy's BIG Wraith (Holds the most gold) as they are easy to access and if you we're to run into the Enemy Jungler, you can simply run back to your side since the walk isn't that far.
Just check the enemy Wraiths when you come to gank mid and take it when you can.

Mid Lane is the easiest to gank, then Top Lane, then Bottom Lane.
It mostly depends on the champ that is going in that lane though.

Mid Lane =
Top Lane =
NOTE: Turret shots do Physical Damage (AD) so armor will reduce the damage.

Bottom Lane =
NOTE: Not all supports will buy a Sightstone, some prefer to buy individual wards.

Amumu's ganking ability is good but don't gank too much or else you will fall behind. Don't stay waiting in a bush for the enemy to get closer when there are camps to clear.

Philosopher's Stone
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Amumu is sorta Mana Hungry since he uses his spells to do damage. This item is mostly bought for the Mana Regeneration that fixes Amumu's mana problem unless you over exessively spam your spells. This also helps you gain more income for your later items and it will give a lot since you buy it early and complete it pretty late. It gives sustain to the sad mummy.

Sunfire Cape

The passive works well with Despair as it increases your overall AOE magic damage per second output. This speeds up your clear time too. If an enemy champion gets damaged by Sunfire Aegis's Passive while you're trying to destroy a turret then you will draw the Turret Aggro. Sunfire Aegis's Passive cannot be turned off.

Abyssal Scepter

Amumu is a close range mage champion. Abyssal Mask's aura range (700 Units Range) is bigger than Curse of the Sad Mummy's range (600 Units Range). All of Amumu's abilities are within range except for Bandage Toss (1,100 Units Range). Remember that Abyssal Mask will also increase Sunfire Aegis's passive damage done to the enemy since it does Magic Damage. It also gives you more Ability Power and Magic Resist and a good amount too which adds to your overall tankyness and damage. (NOTE: This reduces Magic Resist so it can stack with Amumu's Passive Cursed Touch to reduce a total of 45 (Lv 1-6), 50 (Lv 7-12), 45 (Lv 13+) MR. YOU CAN REDUCE BELOW 0 MAGIC RESIST)

Aegis of the Legion/Runic Bulwark

This is like an item that is needed on every team. You don't have to upgrade Aegis of the Legion right after you complete it. The only reason I would ever upgrade Aegis instead of moving on to the next item is if the enemy team composition is AP Heavy or if the AP Champs are being a problem. The aura is what makes this item so good. It's like a core to every team. Just an Aegis of the Legion is enough to make a difference in a teamfight. Without this, your team stays squishier than if they had the bonus, and its more of a disadvantage if the enemy has one built and you don't.

Shurelya's Reverie

This item is mostly bought for it's useful Active. It gives and AOE burst of speed bonus that can be used offensively or defensively. It can be used to initiate or to escape/save a few teammates. The cooldown isn't that long either (60 Seconds) and it builds out of a GP/10 item that helps your early game sustain ( Philosopher's Stone). It also adds to your tanyness with its Health Bonus.

Liandry's Torment

WARNING: You will most likely not finish this item before the end of the game.
This item is mainly bought for it's passive and the Health Bonus. The passive works well with Amumu's Despair as it is constantly on and can't be turned off with any form off CC. Since the Spell is AOE (Multi Target), the duration (3 Sec -> 1.5 Sec) AND damage (5% -> 2.5%) is cut on half WHICH DOES NOT CHANGE THE OVERALL DAMAGE. 5% Current Health Magic Damage over 3 seconds does the same damage over time as 2.5% Current Health Magic Damage over 1.5 seconds. The ONLY difference is that the damage will continue on for 1.5 seconds more. Amumu's Despair does damage every second and Liandry's Torment does damage every half second for 1.5 seconds. This means that as long as the enemy is within range of Amumu's Despair, they will take 0.833% (2.5%/3) Current Health Magic Damage every half second until they get out of range.
NOTE: Sunfire Cape does not trigger this passive.

Mercury Treads VS Mobility Boots
If the enemy has strong CC or Magic Damage then buy Mercury Tread, if not then you can go for Mobility Boots to easily get to the enemy even if you are seen by a ward (only if the enemy is pushed to the turret when you are seen). It gets you where you need to be faster. They are even cheaper but they serve no purpose in combat or a teamfight. It just gets you there. You can get these early but I'd probably sell them late game if manage to build Runic Bulwark
Enchantment: Homeguard VS Enchantment: Captain
Homeguard boots are always useful on ever champion and everytime you recall as it saves time. It gives instant HP and Mana when you recall and even speeds you up to get out there faster. This helps so much with those pesky backdooring champs like Master Yi.
Captain boots are one of the things you would want to go for when you are winning. It helps with initiation since you would want your teammates to help you fight the enemy team when you dive in with Bandage Toss + [[Curse of the Sad Mummy]. This is also useful when you ace the enemy team or somewhat close to that and lead your team to dragon, baron, or to enemy turrets. This is more of an initiation enchantment.

Why get the Enchantment: Captain?
It is your job to initiate by jumping into the enemy team with Bandage Toss and Curse of the Sad Mummy. You would want your teammates to get into the fight faster and help you fight the enemy. The price in my opinion is not too bad but as a jungler, the 20% movement speed for minions won't be that useful since you don't get to hang out with the minions. That 8% movement speed will really help. 8% of lets say 375 movement speed = 30 more movement speed. Not too bad considering that it speeds up your whole team.

As an Initiator, you will be the one to lead the charge to fight the Enemy Team by using Bandage Toss and your ult Curse of the Sad Mummy. DO NOT RISK in a Fail Intiation by using Bandage Toss into Despair and Tantrum as you will be giving the enemy more time to react and cc you leading to a delayed Ultimate.

When intiating, Bandage Toss into Curse of the Sad Mummy on most occasions.
If you Bandage Toss a Squishy Champ like the AD Carry or AP Mage then AUTOATTACK before you ult followed by Despair and Tantrum. If you grab someone other than the AD Carry or AP Mage, ult right after the Bandage Toss, turn on Despair, walk up to the AD Carry or AP Mage, AutoAttack, Tantrum, then follow them for damage over time since they're the highest priority and damage dealer.
NOTE: DON'T GIVE THEM ENOUGH TIME TO CC YOU SO THAT YOUR ULTIMATE IS DELAYED

You have a Game Changing Ultimate. Those 2 Seconds can win you the teamfight as you will be delaying the enemy from responding for a whole 2 Seconds which gives your allies some time to attack without the danger of being killed. (2 Seconds is a lot in the League of Legends World where teamfights last for only a few seconds until one side retreats where its more Chasing than teamfighting)

Ask your Support Champ if they're going to build Runic Bulwark
If they're not, you will have to sacrifice lots of gold and your 5th or 6th item slot for this item (For Ap Tank)
In my opinon, every Team needs a Runic Bulward Late Game.

Some other Options.....
Will of the Ancients = More AP and Spell Vamp, and even your allies benefit from it.
Frozen Heart = More Armor, Mana, and an Attack Speed Slow. Anti AD Carry?
Guardian Angel = Another life? Why not?
Thornmail = Against an AD heavy Team (Buy this only if Frozen Heart is already built on your team, other than that get this item).
Zhonya's Hourglass = More Armor and Ability Power.